Graph f(x)=(3/4)^x

Problem

ƒ(x)=(3/4)x

Solution

1. Identify the type of function. This is an exponential function of the form ƒ(x)=bx where the base b=3/4

2. Determine the behavior. Since 0<3/4<1 the function represents exponential decay, meaning the graph decreases from left to right.

3. Find the y-intercept. Evaluate ƒ(0)=(3/4)0=1 The point is (0,1)

4. Calculate additional points to define the curve.

ƒ*(−1)=(3/4)(−1)=4/3≈1.33

ƒ(1)=(3/4)1=0.75

ƒ(2)=(3/4)2=9/16=0.5625

5. Identify the horizontal asymptote. As x→∞ ƒ(x)→0 The horizontal asymptote is the line y=0

6. Sketch the curve. Plot the points (−1,1.33) (0,1) and (1,0.75) then draw a smooth curve that approaches the x-axis as x increases.

Final Answer

To graph ƒ(x)=(3/4)x plot the y-intercept at (0,1) ensure the curve decreases through points like (−1,4/3) and (1,3/4) and approaches the horizontal asymptote y=0

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